Ramp-up of Hawking Radiation in Bose-Einstein-Condensate Analog Black Holes

Inspired by a recent experiment by Steinhauer and co-workers, we present a simple model which describes the formation of an acoustic black hole in a Bose-Einstein condensate, allowing an analytical computation of the evolution in time of the corresponding density-density correlator. We show the emergence of analog Hawking radiation out of a "quantum atmosphere"region significantly displaced from the horizon. This is quantitatively studied both at T=0 and even in the presence of an initial temperature T, as is always the case experimentally.